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Q30E

Expert-verifiedFound in: Page 1

Book edition
5th

Author(s)
Otto Bretscher

Pages
442 pages

ISBN
9780321796974

**Show that any positive definite matrix A can be written as ****, where B is a positive definite matrix.**

** $A={B}^{2}$ **, where B is a positive definite matrix.

$A={B}^{2}$

Consider the quadratic form $\mathrm{q}\left(\overrightarrow{\mathrm{x}}\right)=\overrightarrow{\mathrm{x}}\xb7\mathrm{A}\overrightarrow{\mathrm{x}}$, where A is a $\mathrm{n}\times \mathrm{n}$ symmetric matrix. If $\mathrm{q}\left(\mathrm{x}\right)$ is positive for all nonzero $\overrightarrow{x}$ in ${\mathrm{\mathbb{R}}}^{n}$, we say A is positive definite. S is an orthogonal matrix that has the following properties:

** ${S}^{-1}AS=D$ **

When D is a diagonal matrix, all of the entries are positive. As a result, A can be written as:

$A=SD{S}^{-1}$

We can write D as a diagonal matrix with positive diagonal elements.

$D={D}_{1}^{2}$

Where is a diagonal matrix with positive diagonal entries, Equation can now be written as follows:

$A={\mathrm{SDS}}^{-1}$

$\Rightarrow A=S\left({D}_{1}^{2}\right){S}^{-1}\phantom{\rule{0ex}{0ex}}\Rightarrow A=S\left({D}_{1}\xb7{D}_{1}\right){S}^{-1}\phantom{\rule{0ex}{0ex}}\Rightarrow A=\left(S{D}_{1}{S}^{-1}\right)\xb7\left(S{D}_{1}{S}^{-1}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow A={B}^{2}$

${\mathrm{SBS}}^{-1}={\mathrm{D}}_{1}$ is a diagonal matrix with positive diagonal elements, hence B is positive definite.

$A={B}^{2}$, where B is a positive definite matrix.

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